-
Notifications
You must be signed in to change notification settings - Fork 0
/
dsa_05_masterthm.html
527 lines (496 loc) · 26.3 KB
/
dsa_05_masterthm.html
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
<!doctype html>
<html lang="en">
<head>
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1.0, maximum-scale=1.0, user-scalable=no">
<link href="css/fontawesome-free-5.15.4-web/css/all.css" rel="stylesheet">
<script src="lib/colorbrewer.v1.min.js" charset="utf-8"></script>
<script src="lib/colorStringStandalone.js" charset="utf-8"></script>
<script type="text/javascript" src="lib/jquery-2.2.4.min.js"></script>
<title>Design & Analysis: Algorithms</title>
<meta name="description" content="CS4851/6851 GSU class">
<meta name="author" content="Sergey M Plis">
<meta name="apple-mobile-web-app-capable" content="yes">
<meta name="apple-mobile-web-app-status-bar-style" content="black-translucent">
<link rel="stylesheet" href="dist/reset.css">
<link rel="stylesheet" href="dist/reveal.css">
<!-- Code syntax highlighting -->
<link rel="stylesheet" href="plugin/highlight/monokai.css" id="highlight-theme">
<!-- <link rel="stylesheet" href="lib/css/zenburn.css"> -->
<link rel="stylesheet" href="css/custom.css">
<link rel="stylesheet" href="dist/theme/aml.css" id="theme">
<!-- Printing and PDF exports -->
<script>
var link = document.createElement( 'link' );
link.rel = 'stylesheet';
link.type = 'text/css';
link.href = window.location.search.match( /print-pdf/gi ) ? 'css/print/pdf.css' : 'css/print/paper.scss';
document.getElementsByTagName( 'head' )[0].appendChild( link );
</script>
</head>
<body>
<div class="reveal">
<!-- In between the <div="reveal"> and the <div class="slides">-->
<!-- <header style="position: absolute; top: 10px; left: 100px; z-index: 500; font-size:100px;background-color: rgba(0,0,0,0); text-align: center !important"></header> -->
<!-- In between the <div="reveal"> and the <div class="slides">-->
<!-- Any section element inside of this container is displayed as a slide -->
<div class="slides">
<section>
<section>
<p>
<h2>Design & Analysis: Algorithms</h2>
<h1>05: Recurrences (master theorem)</h1>
<p>
</section>
<section>
<h3>Outline of the lecture</h3>
<ul>
<li class="fragment roll-in"> Recursion Trees (recap)
<li class="fragment roll-in"> The Master Theorem
<li class="fragment roll-in"> Annihilators
</ul>
</section>
<section>
<h2>Relative Ranking</h2>
<row>
<img style="border:0; box-shadow: 0px 0px 0px
rgba(150, 150, 255, 0.8); width:60%;"
class="reveal" src="figures/competition_time.gif" alt="competition">
<img style="border:0; box-shadow: 0px 0px 0px
rgba(150, 150, 255, 0.8); width:18%;"
class="reveal" src="figures/course_score01.png" alt="competition">
</row>
<div style="margin-top: -20px;">
Send me your private nicknames ASAP
</div>
</section>
</section>
<section>
<section>
<h1>Recursion Trees</h1>
</section>
<section data-background="figures/strassen_tree.svg" data-background-size="contain">
<div id="header-right" style="margin-right: -180px; margin-top: -200px">
<h3 style="margin-left: -1000pt;">Example 2: $T(n) = 3T(n/4) + n^2$</h3>
</div>
<blockquote style="background-color: #eee8d5; width: 100%; font-size: 36px; text-align:left; box-shadow: 5px 5px 10px rgb(51, 51, 51); padding: 10px;" class="left bordered fragment">
<ul>
<li class="fragment roll-in"> We can see that
the i-th level of the tree sums to
$(3/16)^in^2$.
<li class="fragment roll-in"> Further the
depth of the tree is $\log_4 n$ <br> $n/4^d = 1$ implies that $d = \log_4 n$
<li class="fragment roll-in"> So we can see that $T(n) = \sum_{i=0}^{\log_4 n} (3/16)^in^2$
</ul>
</blockquote>
<aside class="notes">
What does each level sum to?
The first (3/16)^1 n^2
The second (3/16)^2 n^2
The next (3/16)^3 n^2
Only then press the button to show itemized list
</aside>
</section>
<section>
<h2>Solution</h2>
\begin{align}
T(n) & \fragment{1}{= \sum_{i=0}^{\log_4 n} (3/16)^in^2} \\
& \fragment{2}{< n^2 \sum_{i=0}^{\infty} (3/16)^i}\\
& \fragment{3}{= \frac{1}{1-{3\over 16}} n^2}\\
& \fragment{4}{ = O(n^2)\\}
\end{align}
</section>
<section data-background="figures/master_theorem_oogway.gif">
</section>
</section>
<section>
<section data-background="figures/master_fighting_a_tree.png" data-background-size="cover" data-vertical-align-top>
<h1 style="text-shadow: 4px 4px 4px #002b36; color: #f1f1f1">Master Theorem</h1>
<div class='slide-footer' style="text-align: left;">
<a href="midjourney.com">midjourney.com</a> Live predatory trees are fighting an old Kung-Fu master in a forest, many trees, light pixar aesthetics
</div>
</section>
<section>
<h1>Master Theorem</h1>
<ul>
<li class="fragment roll-in"> Divide and conquer algorithms often give us running-time recurrences of the form $T(n) = a T(n/b) + f (n)$
<li class="fragment roll-in"> Where $a$ and $b$ are constants and $f(n)$ is some other function.
<li class="fragment roll-in"> The so-called “Master Method” gives us a
general method for solving such recurrences when $f(n)$ is a simple
polynomial.
</ul>
</section>
<section>
<h1>Master Theorem</h1>
<ul>
<li class="fragment roll-in">Unfortunately, the Master Theorem doesn’t work for all functions $f(n)$
<li class="fragment roll-in"> Further many useful recurrences don’t look like $T(n) = a T(n/b) + f (n)$
<li class="fragment roll-in"> However, the theorem allows for very fast solution of recurrences when it applies
</ul>
</section>
<section>
<h1>Master Theorem</h1>
<ul>
<li class="fragment roll-in">Master Theorem is just a special case of the use of recursion
trees
<li class="fragment roll-in">Consider equation $T(n) = a T (n/b) + f (n)$
<li class="fragment roll-in">We start by drawing a recursion tree
</ul>
</section>
<section>
<h2>Recursion Tree</h2>
<ul style="font-size: 28pt;">
<li class="fragment roll-in">The root contains the value $f(n)$
<li class="fragment roll-in">It has a children, each of which contains the value $f(\frac{n}{b})$
<li class="fragment roll-in">Each of these nodes has a children, containing the value
$f(\frac{n}{b^2})$
<li class="fragment roll-in">In general, level $i$ contains $a^i$ nodes with values $f(\frac{n}{b^i})$
<li class="fragment roll-in">Hence the sum of the nodes at the $i^{\mathrm{th}}$ level is $a^i f(\frac{n}{b^i})$
</ul>
</section>
<section>
<h2>Details</h2>
<ul>
<li class="fragment roll-in">The tree stops when we get to the base case for the recurrence
<li class="fragment roll-in">We’ll assume $T(1) = f(1) = \Theta(1)$ is the base case
<li class="fragment roll-in">Thus the depth of the tree is $\log_b n$ and there are $\log_b n + 1$ levels
</ul>
</section>
<section>
<h2>Recursion Tree</h2>
<ul>
<li class="fragment roll-in">Let $T(n)$ be the sum of all values stored in all levels of the
tree:
\begin{align}
T (n) & = f (n)+a f (n/b)+a^2 f (n/b^2)+\\ & \dots+a^i f (n/b^i)+\dots+a^L f (n/b^L)
\end{align}
<li class="fragment roll-in">Where $L = \log_b n$ is the depth of the tree
<li class="fragment roll-in">Since $f(1) = \Theta(1)$, the last term of this summation is $\Theta(a^L) = \Theta(a\log_b n) = \Theta(n\log_b a)$
</ul>
</section>
<section>
<h2>An aside: a "$\log$ fact"</h2>
<ul>
<li class="fragment roll-in">It’s not hard to see that $a^{\log_b n} = n^{\log_b a}$
\begin{align}
a^{\log_b n} & = n^{\log_b a} \mbox{ take }\log_b\\
\log_b a^{\log_b n} & = \log_b n^{\log_b a}\\
\log_b n \log_b a & = \log_b a \log_b n
\end{align}
</ul>
</section>
<section>
<h2>Master Theorem</h2>
<ul>
<li class="fragment roll-in">We can now state the <em>Master Theorem</em>
<li class="fragment roll-in">In a way slightly different from the book
<li class="fragment roll-in">Note:
<blockquote shade style="width:100%;text-align:left; font-size: 34px;" class="fragment" data-fragment-index="1">
The Master Method is just a “short cut” for the recursion tree method. It is less powerful than recursion trees.
</blockquote>
</ul>
</section>
<section>
<h2>Master Method (the theorem)</h2>
<div style="text-align: left;">
The recurrence $T(n) = aT(n/b) + f(n)$ can be solved as follows:
</div>
<ul>
<li class="fragment roll-in">If $a f (n/b) \leq Kf(n)$ for some constant $K < 1$, then <alert>$T(n) = \Theta(f (n))$</alert>.
<li class="fragment roll-in">If $a f (n/b) \geq K f(n)$ for some constant $K > 1$, then <alert>$T(n) = \Theta(n^{\log_b a})$</alert>.
<li class="fragment roll-in">If $a f (n/b) = f (n)$, then <alert>$T(n) = \Theta(f(n) \log_b n)$</alert>.
</ul>
</section>
<section style="text-align:left;">
<div id="header-right" style="margin-right: -80px; margin-top: -50px; font-size: 22pt; color: #268bd2;">
$T (n) = f (n)+a f (n/b)+a^2 f (n/b^2)+ \dots+a^i f (n/b^i)+\dots+a^L f (n/b^L)$
</div>
<h2>Proof (1/3)</h2>
<blockquote style="background-color: #93a1a1; color: #fdf6e3; font-size: 36px; width:100%; text-align:left;" class="fragment roll-in">
If $a f (n/b) \leq Kf(n)$ for some constant $K < 1$, then $T(n) = \Theta(f (n))$.
</blockquote>
<ul>
<li class="fragment roll-in">If $f(n)$ is a constant factor larger than $a f(n/b)$, then the sum is a descending geometric series. The sum of any geometric series is a constant times its largest term. In this case, the largest term is the first term $\Theta(f (n))$.
</ul>
</section>
<section style="text-align:left;">
<div id="header-right" style="margin-right: -80px; margin-top: -50px; font-size: 22pt; color: #268bd2;">
$T (n) = f (n)+a f (n/b)+a^2 f (n/b^2)+ \dots+a^i f (n/b^i)+\dots+a^L f (n/b^L)$
</div>
<h2>Proof (2/3)</h2>
<blockquote style="background-color: #93a1a1; color: #fdf6e3; font-size: 36px; width:100%; text-align:left;" class="fragment roll-in">
If $a f (n/b) \geq K f(n)$ for some constant $K > 1$, then $T(n) = \Theta(n^{\log_b a})$.
</blockquote>
<ul>
<li class="fragment roll-in">If $f(n)$ is a constant factor smaller
than $af(n/b)$, then the sum is an ascending geometric series. The sum
of any geometric series is a constant times its largest term. In this
case, this is the last term, which by our earlier argument is
$\Theta(n^{\log_b a})$.
</ul>
</section>
<section style="text-align:left;">
<div id="header-right" style="margin-right: -80px; margin-top: -50px; font-size: 22pt; color: #268bd2;">
$T (n) = f (n)+a f (n/b)+a^2 f (n/b^2)+ \dots+a^i f (n/b^i)+\dots+a^L f (n/b^L)$
</div>
<h2>Proof (3/3)</h2>
<blockquote style="background-color: #93a1a1; color: #fdf6e3; font-size: 36px; width:100%; text-align:left;" class="fragment roll-in">
If $a f (n/b) = f (n)$, then $T(n) = \Theta(f(n) \log_b n)$.
</blockquote>
<ul>
<li class="fragment roll-in">Finally, if a $f(n/b) = f(n)$, then each of the $L + 1$ terms in
the summation is equal to $f(n)$.
</ul>
</section>
<section>
<h2>Example 1</h2>
<ul>
<li class="fragment roll-in"> $T(n) = T (3n/4) + n$
<li class="fragment roll-in">If we write this as $T (n) = aT (n/b) + f (n)$, then $a = 1$, $b =
4/3$, $f(n) = n$
<li class="fragment roll-in">Here a $f(n/b) = 3n/4$ is smaller than $f (n) = n$ by a factor of $4/3$, so $T(n) = \Theta(n)$
</ul>
</section>
<section>
<h2>Example 2: multiplication</h2>
<ul>
<li class="fragment roll-in">Karatsuba’s multiplication algorithm: $T(n) = 3T (n/2) + n$
<li class="fragment roll-in">If we write this as $T(n) = aT (n/b) + f (n)$, then $a = 3$, $b =
2$, $f(n) = n$
<li class="fragment roll-in">Here $af(n/b) = 3n/2$ is larger than $f (n) = n$ by a factor of
$3/2$, so $T(n) = \Theta(n^{\log_2 3})$
</ul>
<div class="slide-footer">
<a href="https://en.wikipedia.org/wiki/Karatsuba_algorithm">https://en.wikipedia.org/wiki/Karatsuba_algorithm</a>
</div>
</section>
<section>
<h2>Example 3: Mergesort</h2>
<ul>
<li class="fragment roll-in"> Mergesort: $T (n) = 2T (n/2) + n$
<li class="fragment roll-in">If we write this as $T (n) = aT (n/b) + f (n)$, then $a = 2$, $b = 2$, $f (n) = n$
<li class="fragment roll-in">Here a $f(n/b) = f (n)$, so $T (n) = \Theta(n \log n)$
</ul>
</section>
<section>
<h2>Example 4</h2>
<ul>
<li class="fragment roll-in"> $T (n) = T (n/2) + n \log n$
<li class="fragment roll-in">If we write this as $T (n) = aT (n/b) + f (n)$, then $a = 1$, $b = 2$, $f(n) = n \log n$
<li class="fragment roll-in">Here a $f (n/b) = n/2 \log n/2$ is smaller than $f (n) = n \log n$ by
a constant factor, so $T (n) = \Theta(n \log n)$
</ul>
</section>
<section>
<h2>In Class Exercise 1 <img style="border:0; box-shadow: 0px 0px 0px rgba(150, 150, 255, 1);" width="100"
src="figures/dolphin_swim.webp" alt="Cormen Algs">
</h2>
<ul>
<li class="fragment roll-in">Consider the recurrence: $T (n) = 4T (n/2) + n \log n$
<li class="fragment roll-in">Q: What is $f (n)$ and $a f (n/b)$?
<li class="fragment roll-in">Q: Which of the three cases does the recurrence fall under
(when n is large)?
<li class="fragment roll-in">Q: What is the solution to this recurrence?
</ul>
</section>
<section>
<h2>Solution</h2>
\begin{align}
4\frac{n}{2} \log \frac{n}{2} & \quad ? \quad n \log n\\
2 n \log \frac{n}{2} & \quad ? \quad n \log n\\
2n\log n - 2n\log 2 & \quad ? \quad n \log n\\
2 - \frac{2}{\log n} & \quad ? \quad 1\\
2 (1 - \frac{1}{\log n} ) & \geq 1 \text{ case 2 } \Theta(n^2)\\
\end{align}
</section>
<section>
<h2>In Class Exercise 2 <img style="border:0; box-shadow: 0px 0px 0px rgba(150, 150, 255, 1);" width="100"
src="figures/dolphin_swim.webp" alt="Cormen Algs">
</h2>
<ul>
<li class="fragment roll-in">Consider the recurrence: $T (n) = 2T (n/4) + n \log n$
<li class="fragment roll-in">Q: What is $f (n)$ and $a f (n/b)$?
<li class="fragment roll-in">Q: Which of the three cases does the recurrence fall under
(when n is large)?
<li class="fragment roll-in">Q: What is the solution to this recurrence?
</ul>
</section>
<section>
<h2>Solution</h2>
\begin{align}
\frac{n}{2} \log \frac{n}{4} & \quad ? \quad n \log n\\
\frac{n}{2} \log n - \frac{n}{2} \log 4 & \quad ? \quad n \log n\\
\frac{1}{2}\log n - \frac{1}{\log n} & \quad ? \quad 1\\
\frac{1}{2} (1 - \frac{2}{\log n} ) & \geq 1 \text{ case 1 } \Theta(n \log n)\\
\end{align}
</section>
<section>
<h2>Take Away</h2>
<ul>
<li class="fragment roll-in">Recursion tree and Master method are good tools for solving
many recurrences
<li class="fragment roll-in">However these methods are limited (they can’t help us get
guesses for recurrences like $f (n) = f (n − 1) + f (n − 2))$
<li class="fragment roll-in">For info on how to solve these other more difficult recurrences, review the notes on annihilators by Jeff Erikson <a href="https://jeffe.cs.illinois.edu/teaching/algorithms/notes/99-recurrences.pdf">in this appendix</a>.
</ul>
</section>
<section>
<h2>Assigned reading</h2>
<row>
<col60>
<img style="border:0; box-shadow: 0px 0px 0px rgba(150, 150, 255, 1);" width="80%"
src="figures/cormen_algs.jpeg" alt="Cormen Algs">
</col60>
<col40>
Chapter 4: Divide and Conquer<p>
</col40>
</row>
</section>
</section>
<section>
<h2>See you</h2>
Monday January 30th
</section>
</div>
</div>
<script src="dist/reveal.js"></script>
<!-- <link rel="stylesheet" href="lib/css/monokai.css"> -->
<script src="plugin/highlight/highlight.js"></script>
<script src="plugin/math/math.js"></script>
<script src="plugin/chalkboard/plugin.js"></script>
<script src="plugin/notes/notes.js"></script>
<script src="plugin/zoom/zoom.js"></script>
<script src="plugin/menu/menu.js"></script>
<script>
// Full list of configuration options available at:
// https://github.com/hakimel/reveal.js#configuration
let notes = document.querySelectorAll('aside.notes');
notes.forEach(n => {
let html = n.innerHTML;
html = html.trim().replace(/\n/g, '<br/>');
n.innerHTML = html;
});
Reveal.initialize({
// history: true,
hash: true,
margin: 0.01,
minScale: 0.01,
maxScale: 1.23,
menu: {
themes: true,
openSlideNumber: true,
openButton: false,
},
customcontrols: {
slideNumberCSS : 'position: fixed; display: block; right: 90px; top: auto; left: auto; width: 50px; bottom: 30px; z-index: 31; font-family: Helvetica, sans-serif; font-size: 12px; line-height: 1; padding: 5px; text-align: center; border-radius: 10px; background-color: rgba(128,128,128,.5)',
controls: [
{ icon: '<i class="fa fa-caret-left"></i>',
css: 'position: fixed; right: 60px; bottom: 30px; z-index: 30; font-size: 24px;',
action: 'Reveal.prev(); return false;'
},
{ icon: '<i class="fa fa-caret-right"></i>',
css: 'position: fixed; right: 30px; bottom: 30px; z-index: 30; font-size: 24px;',
action: 'Reveal.next(); return false;'
}
]
},
chalkboard: {
boardmarkerWidth: 1,
chalkWidth: 2,
chalkEffect: 1,
slideWidth: Reveal.width,
slideHeight: Reveal.height,
toggleNotesButton: false,
toggleChalkboardButton: false,
//src: "chalkboards/chalkboard_em2.json",
readOnly: false,
theme: "blackboard",
eraser: { src: "plugin/chalkboard/img/sponge.png", radius: 30},
},
math: {
mathjax: 'https://cdn.jsdelivr.net/gh/mathjax/[email protected]/MathJax.js',
config: 'TeX-AMS_SVG-full',
// pass other options into `MathJax.Hub.Config()`
TeX: {
Macros: {
RR: '\\mathbb{R}',
PP: '\\mathbb{P}',
EE: '\\mathbb{E}',
NN: '\\mathbb{N}',
vth: '\\vec{\\theta}',
loss: '{\\cal l}',
hclass: '{\\cal H}',
CD: '{\\cal D}',
def: '\\stackrel{\\text{def}}{=}',
pag: ['\\text{pa}_{{\cal G}^{#1}}(#2)}', 2],
vec: ['\\boldsymbol{\\mathbf #1}', 1],
set: [ '\\left\\{#1 \\; : \\; #2\\right\\}', 2 ],
bm: ['\\boldsymbol{\\mathbf #1}', 1],
argmin: ['\\operatorname\{arg\\,min\\,\}'],
argmax: ['\\operatorname\{arg\\,max\\,\}'],
prob: ["\\mbox{#1$\\left(#2\\right)$}", 2],
},
loader: {load: ['[tex]/color']},
extensions: ["color.js"],
tex: {packages: {'[+]': ['color']}},
svg: {
fontCache: 'global'
}
}
},
plugins: [ RevealMath, RevealChalkboard, RevealHighlight, RevealNotes, RevealZoom, RevealMenu ],
});
Reveal.configure({ fragments: true }); // set false when developing to see everything at once
Reveal.configure({ slideNumber: true });
//Reveal.configure({ history: true });
Reveal.configure({ slideNumber: 'c / t' });
Reveal.addEventListener( 'darkside', function() {
document.getElementById('theme').setAttribute('href','dist/theme/aml_dark.css');
}, false );
Reveal.addEventListener( 'brightside', function() {
document.getElementById('theme').setAttribute('href','dist/theme/aml.css');
}, false );
</script>
<style type="text/css">
/* 1. Style header/footer <div> so they are positioned as desired. */
#header-left {
position: absolute;
top: 0%;
left: 0%;
}
#header-right {
position: absolute;
top: 0%;
right: 0%;
}
#footer-left {
position: absolute;
bottom: 0%;
left: 0%;
}
</style>
<!-- // 2. Create hidden header/footer -->
<div id="hidden" style="background; display:none;">
<div id="header">
<div id="header-left"><h4>CS4520</h4></div>
<div id="header-right"><h4>Algorithms</h4></div>
<div id="footer-left">
<img style="border:0; box-shadow: 0px 0px 0px rgba(150, 150, 255, 1);" width="100"
src="figures/flowchart.png" alt="robot learning">
</div>
</div>
</div>
<script type="text/javascript">
// 3. On Reveal.js ready event, copy header/footer <div> into each `.slide-background` <div>
var header = $('#header').html();
if ( window.location.search.match( /print-pdf/gi ) ) {
Reveal.addEventListener( 'ready', function( event ) {
$('.slide-background').append(header);
});
}
else {
$('div.reveal').append(header);
}
</script>
</body>
</html>